Start taking pictures of everything.
Make sure you save all the emails regarding this situation that you have sent to the landlord.
Ask your friends if they are willing to talk in front of a camera about the situation and their experience.
After you have all that I advise you to call Shelter England for further advice on what to do. Most likely they will tell you to contact the council and explain that you are living in an unsanitary condition and that you are living in a health hazard.
Additionally, if you start feeling sick, stressed or physically hurt due to these conditions go to a doctor and make sure you save the prescriptions and reasons you are going for.
This will help build a strong case against your landlord and if you ultimately HAVE to go to court you can win a large sum of money. The least you can do is tell the council which will come and inspect your property. After that, they will decide if your landlord has to carry out repairs and has to take care of the roaches and rats. You can also demand to be let go on the contract due to the conditions but likely the landlord will not let you because if they do then everyone will follow you.